rsin B (should all be same)

Percentage Accurate: 76.7% → 99.5%

Time: 16.2s

Alternatives: 9

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ r \cdot \frac{\sin b}{\cos \left(a + b\right)} \end{array} \]

FPCore

(FPCore (r a b) :precision binary64 (* r (/ (sin b) (cos (+ a b)))))

C

double code(double r, double a, double b) {
	return r * (sin(b) / cos((a + b)));
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = r * (sin(b) / cos((a + b)))
end function

Java

public static double code(double r, double a, double b) {
	return r * (Math.sin(b) / Math.cos((a + b)));
}

Python

def code(r, a, b):
	return r * (math.sin(b) / math.cos((a + b)))

Julia

function code(r, a, b)
	return Float64(r * Float64(sin(b) / cos(Float64(a + b))))
end

MATLAB

function tmp = code(r, a, b)
	tmp = r * (sin(b) / cos((a + b)));
end

Wolfram

code[r_, a_, b_] := N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 76.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ r \cdot \frac{\sin b}{\cos \left(a + b\right)} \end{array} \]

FPCore

(FPCore (r a b) :precision binary64 (* r (/ (sin b) (cos (+ a b)))))

C

double code(double r, double a, double b) {
	return r * (sin(b) / cos((a + b)));
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = r * (sin(b) / cos((a + b)))
end function

Java

public static double code(double r, double a, double b) {
	return r * (Math.sin(b) / Math.cos((a + b)));
}

Python

def code(r, a, b):
	return r * (math.sin(b) / math.cos((a + b)))

Julia

function code(r, a, b)
	return Float64(r * Float64(sin(b) / cos(Float64(a + b))))
end

MATLAB

function tmp = code(r, a, b)
	tmp = r * (sin(b) / cos((a + b)));
end

Wolfram

code[r_, a_, b_] := N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, \sin b \cdot \left(-\sin a\right)\right)} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (* r (/ (sin b) (fma (cos b) (cos a) (* (sin b) (- (sin a)))))))

C

double code(double r, double a, double b) {
	return r * (sin(b) / fma(cos(b), cos(a), (sin(b) * -sin(a))));
}

Julia

function code(r, a, b)
	return Float64(r * Float64(sin(b) / fma(cos(b), cos(a), Float64(sin(b) * Float64(-sin(a))))))
end

Wolfram

code[r_, a_, b_] := N[(r * N[(N[Sin[b], $MachinePrecision] / N[(N[Cos[b], $MachinePrecision] * N[Cos[a], $MachinePrecision] + N[(N[Sin[b], $MachinePrecision] * (-N[Sin[a], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.9%

   \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
2. Step-by-step derivation

   1. +-commutative 78.9%

      \[\leadsto r \cdot \frac{\sin b}{\cos \color{blue}{\left(b + a\right)}} \]

3. Simplified 78.9%

   \[\leadsto \color{blue}{r \cdot \frac{\sin b}{\cos \left(b + a\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. cos-sum 99.5%

      \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

   2. cancel-sign-sub-inv 99.5%

      \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a + \left(-\sin b\right) \cdot \sin a}} \]

   3. fma-define 99.5%

      \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \sin a\right)}} \]

6. Applied egg-rr 99.5%

   \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \sin a\right)}} \]

7. Final simplification 99.5%

   \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, \sin b \cdot \left(-\sin a\right)\right)} \]
8. Add Preprocessing

Alternative 2: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ r \cdot \frac{\sin b}{\cos b \cdot \cos a - \sin b \cdot \sin a} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (* r (/ (sin b) (- (* (cos b) (cos a)) (* (sin b) (sin a))))))

C

double code(double r, double a, double b) {
	return r * (sin(b) / ((cos(b) * cos(a)) - (sin(b) * sin(a))));
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = r * (sin(b) / ((cos(b) * cos(a)) - (sin(b) * sin(a))))
end function

Java

public static double code(double r, double a, double b) {
	return r * (Math.sin(b) / ((Math.cos(b) * Math.cos(a)) - (Math.sin(b) * Math.sin(a))));
}

Python

def code(r, a, b):
	return r * (math.sin(b) / ((math.cos(b) * math.cos(a)) - (math.sin(b) * math.sin(a))))

Julia

function code(r, a, b)
	return Float64(r * Float64(sin(b) / Float64(Float64(cos(b) * cos(a)) - Float64(sin(b) * sin(a)))))
end

MATLAB

function tmp = code(r, a, b)
	tmp = r * (sin(b) / ((cos(b) * cos(a)) - (sin(b) * sin(a))));
end

Wolfram

code[r_, a_, b_] := N[(r * N[(N[Sin[b], $MachinePrecision] / N[(N[(N[Cos[b], $MachinePrecision] * N[Cos[a], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[b], $MachinePrecision] * N[Sin[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.9%

   \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
2. Step-by-step derivation

   1. +-commutative 78.9%

      \[\leadsto r \cdot \frac{\sin b}{\cos \color{blue}{\left(b + a\right)}} \]

3. Simplified 78.9%

   \[\leadsto \color{blue}{r \cdot \frac{\sin b}{\cos \left(b + a\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. cos-sum 99.5%

      \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

6. Applied egg-rr 99.5%

   \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

7. Final simplification 99.5%

   \[\leadsto r \cdot \frac{\sin b}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]
8. Add Preprocessing

Alternative 3: 76.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.5 \cdot 10^{-5} \lor \neg \left(a \leq 8.5 \cdot 10^{-6}\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= a -3.5e-5) (not (<= a 8.5e-6)))
   (* r (/ (sin b) (cos a)))
   (* r (tan b))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((a <= -3.5e-5) || !(a <= 8.5e-6)) {
		tmp = r * (sin(b) / cos(a));
	} else {
		tmp = r * tan(b);
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((a <= (-3.5d-5)) .or. (.not. (a <= 8.5d-6))) then
        tmp = r * (sin(b) / cos(a))
    else
        tmp = r * tan(b)
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((a <= -3.5e-5) || !(a <= 8.5e-6)) {
		tmp = r * (Math.sin(b) / Math.cos(a));
	} else {
		tmp = r * Math.tan(b);
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (a <= -3.5e-5) or not (a <= 8.5e-6):
		tmp = r * (math.sin(b) / math.cos(a))
	else:
		tmp = r * math.tan(b)
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((a <= -3.5e-5) || !(a <= 8.5e-6))
		tmp = Float64(r * Float64(sin(b) / cos(a)));
	else
		tmp = Float64(r * tan(b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((a <= -3.5e-5) || ~((a <= 8.5e-6)))
		tmp = r * (sin(b) / cos(a));
	else
		tmp = r * tan(b);
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[a, -3.5e-5], N[Not[LessEqual[a, 8.5e-6]], $MachinePrecision]], N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.4999999999999997e-5 or 8.4999999999999999e-6 < a

   1. Initial program 56.8%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. +-commutative 56.8%

         \[\leadsto r \cdot \frac{\sin b}{\cos \color{blue}{\left(b + a\right)}} \]

   3. Simplified 56.8%

      \[\leadsto \color{blue}{r \cdot \frac{\sin b}{\cos \left(b + a\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around 0 57.1%

      \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos a}} \]

   if -3.4999999999999997e-5 < a < 8.4999999999999999e-6

   1. Initial program 99.0%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. +-commutative 99.0%

         \[\leadsto r \cdot \frac{\sin b}{\cos \color{blue}{\left(b + a\right)}} \]

   3. Simplified 99.0%

      \[\leadsto \color{blue}{r \cdot \frac{\sin b}{\cos \left(b + a\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. cos-sum 99.7%

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      2. cancel-sign-sub-inv 99.7%

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a + \left(-\sin b\right) \cdot \sin a}} \]

      3. fma-define 99.7%

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \sin a\right)}} \]

   6. Applied egg-rr 99.7%

      \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \sin a\right)}} \]
   7. Step-by-step derivation

      1. frac-2neg 99.7%

         \[\leadsto r \cdot \color{blue}{\frac{-\sin b}{-\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \sin a\right)}} \]

      2. neg-sub0 99.7%

         \[\leadsto r \cdot \frac{\color{blue}{0 - \sin b}}{-\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \sin a\right)} \]

      3. div-sub 99.7%

         \[\leadsto r \cdot \color{blue}{\left(\frac{0}{-\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \sin a\right)} - \frac{\sin b}{-\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \sin a\right)}\right)} \]

      4. distribute-lft-neg-out 99.7%

         \[\leadsto r \cdot \left(\frac{0}{-\mathsf{fma}\left(\cos b, \cos a, \color{blue}{-\sin b \cdot \sin a}\right)} - \frac{\sin b}{-\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \sin a\right)}\right) \]

      5. fma-neg 99.7%

         \[\leadsto r \cdot \left(\frac{0}{-\color{blue}{\left(\cos b \cdot \cos a - \sin b \cdot \sin a\right)}} - \frac{\sin b}{-\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \sin a\right)}\right) \]

      6. cos-sum 99.7%

         \[\leadsto r \cdot \left(\frac{0}{-\color{blue}{\cos \left(b + a\right)}} - \frac{\sin b}{-\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \sin a\right)}\right) \]

      7. add-sqr-sqrt 50.3%

         \[\leadsto r \cdot \left(\frac{0}{-\cos \left(b + a\right)} - \frac{\color{blue}{\sqrt{\sin b} \cdot \sqrt{\sin b}}}{-\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \sin a\right)}\right) \]

      8. sqrt-unprod 54.0%

         \[\leadsto r \cdot \left(\frac{0}{-\cos \left(b + a\right)} - \frac{\color{blue}{\sqrt{\sin b \cdot \sin b}}}{-\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \sin a\right)}\right) \]

      9. sqr-neg 54.0%

         \[\leadsto r \cdot \left(\frac{0}{-\cos \left(b + a\right)} - \frac{\sqrt{\color{blue}{\left(-\sin b\right) \cdot \left(-\sin b\right)}}}{-\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \sin a\right)}\right) \]

      10. sqrt-unprod 7.8%

          \[\leadsto r \cdot \left(\frac{0}{-\cos \left(b + a\right)} - \frac{\color{blue}{\sqrt{-\sin b} \cdot \sqrt{-\sin b}}}{-\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \sin a\right)}\right) \]

      11. add-sqr-sqrt 15.6%

          \[\leadsto r \cdot \left(\frac{0}{-\cos \left(b + a\right)} - \frac{\color{blue}{-\sin b}}{-\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \sin a\right)}\right) \]

      12. distribute-lft-neg-out 15.6%

          \[\leadsto r \cdot \left(\frac{0}{-\cos \left(b + a\right)} - \frac{-\sin b}{-\mathsf{fma}\left(\cos b, \cos a, \color{blue}{-\sin b \cdot \sin a}\right)}\right) \]

   8. Applied egg-rr 15.6%

      \[\leadsto r \cdot \color{blue}{\left(\frac{0}{-\cos \left(b + a\right)} - \frac{\sin b}{\cos \left(b + a\right)}\right)} \]
   9. Step-by-step derivation

      1. div0 15.6%

         \[\leadsto r \cdot \left(\color{blue}{0} - \frac{\sin b}{\cos \left(b + a\right)}\right) \]

      2. neg-sub0 15.6%

         \[\leadsto r \cdot \color{blue}{\left(-\frac{\sin b}{\cos \left(b + a\right)}\right)} \]

      3. distribute-neg-frac 15.6%

         \[\leadsto r \cdot \color{blue}{\frac{-\sin b}{\cos \left(b + a\right)}} \]

   10. Simplified 15.6%

       \[\leadsto r \cdot \color{blue}{\frac{-\sin b}{\cos \left(b + a\right)}} \]

   11. Taylor expanded in a around 0 15.6%

       \[\leadsto \color{blue}{-1 \cdot \frac{r \cdot \sin b}{\cos b}} \]
   12. Step-by-step derivation

       1. frac-2neg 15.6%

          \[\leadsto -1 \cdot \color{blue}{\frac{-r \cdot \sin b}{-\cos b}} \]

       2. distribute-frac-neg 15.6%

          \[\leadsto -1 \cdot \color{blue}{\left(-\frac{r \cdot \sin b}{-\cos b}\right)} \]

       3. add-sqr-sqrt 7.8%

          \[\leadsto -1 \cdot \left(-\frac{r \cdot \color{blue}{\left(\sqrt{\sin b} \cdot \sqrt{\sin b}\right)}}{-\cos b}\right) \]

       4. sqrt-unprod 46.8%

          \[\leadsto -1 \cdot \left(-\frac{r \cdot \color{blue}{\sqrt{\sin b \cdot \sin b}}}{-\cos b}\right) \]

       5. sqr-neg 46.8%

          \[\leadsto -1 \cdot \left(-\frac{r \cdot \sqrt{\color{blue}{\left(-\sin b\right) \cdot \left(-\sin b\right)}}}{-\cos b}\right) \]

       6. sqrt-unprod 48.7%

          \[\leadsto -1 \cdot \left(-\frac{r \cdot \color{blue}{\left(\sqrt{-\sin b} \cdot \sqrt{-\sin b}\right)}}{-\cos b}\right) \]

       7. add-sqr-sqrt 99.0%

          \[\leadsto -1 \cdot \left(-\frac{r \cdot \color{blue}{\left(-\sin b\right)}}{-\cos b}\right) \]

       8. distribute-rgt-neg-in 99.0%

          \[\leadsto -1 \cdot \left(-\frac{\color{blue}{-r \cdot \sin b}}{-\cos b}\right) \]

       9. frac-2neg 99.0%

          \[\leadsto -1 \cdot \left(-\color{blue}{\frac{r \cdot \sin b}{\cos b}}\right) \]

       10. associate-/l* 99.0%

           \[\leadsto -1 \cdot \left(-\color{blue}{r \cdot \frac{\sin b}{\cos b}}\right) \]

       11. quot-tan 99.1%

           \[\leadsto -1 \cdot \left(-r \cdot \color{blue}{\tan b}\right) \]

   13. Applied egg-rr 99.1%

       \[\leadsto -1 \cdot \color{blue}{\left(-r \cdot \tan b\right)} \]
   14. Step-by-step derivation

       1. distribute-rgt-neg-in 99.1%

          \[\leadsto -1 \cdot \color{blue}{\left(r \cdot \left(-\tan b\right)\right)} \]

   15. Simplified 99.1%

       \[\leadsto -1 \cdot \color{blue}{\left(r \cdot \left(-\tan b\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.5 \cdot 10^{-5} \lor \neg \left(a \leq 8.5 \cdot 10^{-6}\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ r \cdot \frac{\sin b}{\cos \left(b + a\right)} \end{array} \]

FPCore

(FPCore (r a b) :precision binary64 (* r (/ (sin b) (cos (+ b a)))))

C

double code(double r, double a, double b) {
	return r * (sin(b) / cos((b + a)));
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = r * (sin(b) / cos((b + a)))
end function

Java

public static double code(double r, double a, double b) {
	return r * (Math.sin(b) / Math.cos((b + a)));
}

Python

def code(r, a, b):
	return r * (math.sin(b) / math.cos((b + a)))

Julia

function code(r, a, b)
	return Float64(r * Float64(sin(b) / cos(Float64(b + a))))
end

MATLAB

function tmp = code(r, a, b)
	tmp = r * (sin(b) / cos((b + a)));
end

Wolfram

code[r_, a_, b_] := N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.9%

   \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
2. Add Preprocessing

3. Final simplification 78.9%

   \[\leadsto r \cdot \frac{\sin b}{\cos \left(b + a\right)} \]
4. Add Preprocessing

Alternative 5: 76.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sin b \cdot \frac{r}{\cos \left(b + a\right)} \end{array} \]

FPCore

(FPCore (r a b) :precision binary64 (* (sin b) (/ r (cos (+ b a)))))

C

double code(double r, double a, double b) {
	return sin(b) * (r / cos((b + a)));
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = sin(b) * (r / cos((b + a)))
end function

Java

public static double code(double r, double a, double b) {
	return Math.sin(b) * (r / Math.cos((b + a)));
}

Python

def code(r, a, b):
	return math.sin(b) * (r / math.cos((b + a)))

Julia

function code(r, a, b)
	return Float64(sin(b) * Float64(r / cos(Float64(b + a))))
end

MATLAB

function tmp = code(r, a, b)
	tmp = sin(b) * (r / cos((b + a)));
end

Wolfram

code[r_, a_, b_] := N[(N[Sin[b], $MachinePrecision] * N[(r / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.9%

   \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
2. Step-by-step derivation

   1. associate-*r/ 78.9%

      \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos \left(a + b\right)}} \]

   2. +-commutative 78.9%

      \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(b + a\right)}} \]

3. Simplified 78.9%

   \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos \left(b + a\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. *-commutative 78.9%

      \[\leadsto \frac{\color{blue}{\sin b \cdot r}}{\cos \left(b + a\right)} \]

   2. associate-/l* 78.9%

      \[\leadsto \color{blue}{\sin b \cdot \frac{r}{\cos \left(b + a\right)}} \]

6. Applied egg-rr 78.9%

   \[\leadsto \color{blue}{\sin b \cdot \frac{r}{\cos \left(b + a\right)}} \]

7. Final simplification 78.9%

   \[\leadsto \sin b \cdot \frac{r}{\cos \left(b + a\right)} \]
8. Add Preprocessing

Alternative 6: 76.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{r \cdot \sin b}{\cos \left(b + a\right)} \end{array} \]

FPCore

(FPCore (r a b) :precision binary64 (/ (* r (sin b)) (cos (+ b a))))

C

double code(double r, double a, double b) {
	return (r * sin(b)) / cos((b + a));
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (r * sin(b)) / cos((b + a))
end function

Java

public static double code(double r, double a, double b) {
	return (r * Math.sin(b)) / Math.cos((b + a));
}

Python

def code(r, a, b):
	return (r * math.sin(b)) / math.cos((b + a))

Julia

function code(r, a, b)
	return Float64(Float64(r * sin(b)) / cos(Float64(b + a)))
end

MATLAB

function tmp = code(r, a, b)
	tmp = (r * sin(b)) / cos((b + a));
end

Wolfram

code[r_, a_, b_] := N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[Cos[N[(b + a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.9%

   \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
2. Step-by-step derivation

   1. associate-*r/ 78.9%

      \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos \left(a + b\right)}} \]

   2. +-commutative 78.9%

      \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(b + a\right)}} \]

3. Simplified 78.9%

   \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos \left(b + a\right)}} \]
4. Add Preprocessing

5. Final simplification 78.9%

   \[\leadsto \frac{r \cdot \sin b}{\cos \left(b + a\right)} \]
6. Add Preprocessing

Alternative 7: 75.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.55 \lor \neg \left(b \leq 1.02 \cdot 10^{-35}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \end{array} \]

FPCore

(FPCore (r a b)
 :precision binary64
 (if (or (<= b -0.55) (not (<= b 1.02e-35)))
   (* r (tan b))
   (* b (/ r (cos a)))))

C

double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.55) || !(b <= 1.02e-35)) {
		tmp = r * tan(b);
	} else {
		tmp = b * (r / cos(a));
	}
	return tmp;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((b <= (-0.55d0)) .or. (.not. (b <= 1.02d-35))) then
        tmp = r * tan(b)
    else
        tmp = b * (r / cos(a))
    end if
    code = tmp
end function

Java

public static double code(double r, double a, double b) {
	double tmp;
	if ((b <= -0.55) || !(b <= 1.02e-35)) {
		tmp = r * Math.tan(b);
	} else {
		tmp = b * (r / Math.cos(a));
	}
	return tmp;
}

Python

def code(r, a, b):
	tmp = 0
	if (b <= -0.55) or not (b <= 1.02e-35):
		tmp = r * math.tan(b)
	else:
		tmp = b * (r / math.cos(a))
	return tmp

Julia

function code(r, a, b)
	tmp = 0.0
	if ((b <= -0.55) || !(b <= 1.02e-35))
		tmp = Float64(r * tan(b));
	else
		tmp = Float64(b * Float64(r / cos(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(r, a, b)
	tmp = 0.0;
	if ((b <= -0.55) || ~((b <= 1.02e-35)))
		tmp = r * tan(b);
	else
		tmp = b * (r / cos(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[r_, a_, b_] := If[Or[LessEqual[b, -0.55], N[Not[LessEqual[b, 1.02e-35]], $MachinePrecision]], N[(r * N[Tan[b], $MachinePrecision]), $MachinePrecision], N[(b * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -0.55000000000000004 or 1.01999999999999995e-35 < b

   1. Initial program 59.6%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. +-commutative 59.6%

         \[\leadsto r \cdot \frac{\sin b}{\cos \color{blue}{\left(b + a\right)}} \]

   3. Simplified 59.6%

      \[\leadsto \color{blue}{r \cdot \frac{\sin b}{\cos \left(b + a\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. cos-sum 99.2%

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      2. cancel-sign-sub-inv 99.2%

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a + \left(-\sin b\right) \cdot \sin a}} \]

      3. fma-define 99.3%

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \sin a\right)}} \]

   6. Applied egg-rr 99.3%

      \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \sin a\right)}} \]
   7. Step-by-step derivation

      1. frac-2neg 99.3%

         \[\leadsto r \cdot \color{blue}{\frac{-\sin b}{-\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \sin a\right)}} \]

      2. neg-sub0 99.3%

         \[\leadsto r \cdot \frac{\color{blue}{0 - \sin b}}{-\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \sin a\right)} \]

      3. div-sub 99.3%

         \[\leadsto r \cdot \color{blue}{\left(\frac{0}{-\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \sin a\right)} - \frac{\sin b}{-\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \sin a\right)}\right)} \]

      4. distribute-lft-neg-out 99.3%

         \[\leadsto r \cdot \left(\frac{0}{-\mathsf{fma}\left(\cos b, \cos a, \color{blue}{-\sin b \cdot \sin a}\right)} - \frac{\sin b}{-\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \sin a\right)}\right) \]

      5. fma-neg 99.3%

         \[\leadsto r \cdot \left(\frac{0}{-\color{blue}{\left(\cos b \cdot \cos a - \sin b \cdot \sin a\right)}} - \frac{\sin b}{-\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \sin a\right)}\right) \]

      6. cos-sum 99.3%

         \[\leadsto r \cdot \left(\frac{0}{-\color{blue}{\cos \left(b + a\right)}} - \frac{\sin b}{-\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \sin a\right)}\right) \]

      7. add-sqr-sqrt 54.3%

         \[\leadsto r \cdot \left(\frac{0}{-\cos \left(b + a\right)} - \frac{\color{blue}{\sqrt{\sin b} \cdot \sqrt{\sin b}}}{-\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \sin a\right)}\right) \]

      8. sqrt-unprod 55.5%

         \[\leadsto r \cdot \left(\frac{0}{-\cos \left(b + a\right)} - \frac{\color{blue}{\sqrt{\sin b \cdot \sin b}}}{-\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \sin a\right)}\right) \]

      9. sqr-neg 55.5%

         \[\leadsto r \cdot \left(\frac{0}{-\cos \left(b + a\right)} - \frac{\sqrt{\color{blue}{\left(-\sin b\right) \cdot \left(-\sin b\right)}}}{-\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \sin a\right)}\right) \]

      10. sqrt-unprod 0.9%

          \[\leadsto r \cdot \left(\frac{0}{-\cos \left(b + a\right)} - \frac{\color{blue}{\sqrt{-\sin b} \cdot \sqrt{-\sin b}}}{-\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \sin a\right)}\right) \]

      11. add-sqr-sqrt 2.1%

          \[\leadsto r \cdot \left(\frac{0}{-\cos \left(b + a\right)} - \frac{\color{blue}{-\sin b}}{-\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \sin a\right)}\right) \]

      12. distribute-lft-neg-out 2.1%

          \[\leadsto r \cdot \left(\frac{0}{-\cos \left(b + a\right)} - \frac{-\sin b}{-\mathsf{fma}\left(\cos b, \cos a, \color{blue}{-\sin b \cdot \sin a}\right)}\right) \]

   8. Applied egg-rr 5.9%

      \[\leadsto r \cdot \color{blue}{\left(\frac{0}{-\cos \left(b + a\right)} - \frac{\sin b}{\cos \left(b + a\right)}\right)} \]
   9. Step-by-step derivation

      1. div0 5.9%

         \[\leadsto r \cdot \left(\color{blue}{0} - \frac{\sin b}{\cos \left(b + a\right)}\right) \]

      2. neg-sub0 5.9%

         \[\leadsto r \cdot \color{blue}{\left(-\frac{\sin b}{\cos \left(b + a\right)}\right)} \]

      3. distribute-neg-frac 5.9%

         \[\leadsto r \cdot \color{blue}{\frac{-\sin b}{\cos \left(b + a\right)}} \]

   10. Simplified 5.9%

       \[\leadsto r \cdot \color{blue}{\frac{-\sin b}{\cos \left(b + a\right)}} \]

   11. Taylor expanded in a around 0 6.5%

       \[\leadsto \color{blue}{-1 \cdot \frac{r \cdot \sin b}{\cos b}} \]
   12. Step-by-step derivation

       1. frac-2neg 6.5%

          \[\leadsto -1 \cdot \color{blue}{\frac{-r \cdot \sin b}{-\cos b}} \]

       2. distribute-frac-neg 6.5%

          \[\leadsto -1 \cdot \color{blue}{\left(-\frac{r \cdot \sin b}{-\cos b}\right)} \]

       3. add-sqr-sqrt 3.6%

          \[\leadsto -1 \cdot \left(-\frac{r \cdot \color{blue}{\left(\sqrt{\sin b} \cdot \sqrt{\sin b}\right)}}{-\cos b}\right) \]

       4. sqrt-unprod 30.1%

          \[\leadsto -1 \cdot \left(-\frac{r \cdot \color{blue}{\sqrt{\sin b \cdot \sin b}}}{-\cos b}\right) \]

       5. sqr-neg 30.1%

          \[\leadsto -1 \cdot \left(-\frac{r \cdot \sqrt{\color{blue}{\left(-\sin b\right) \cdot \left(-\sin b\right)}}}{-\cos b}\right) \]

       6. sqrt-unprod 26.5%

          \[\leadsto -1 \cdot \left(-\frac{r \cdot \color{blue}{\left(\sqrt{-\sin b} \cdot \sqrt{-\sin b}\right)}}{-\cos b}\right) \]

       7. add-sqr-sqrt 58.9%

          \[\leadsto -1 \cdot \left(-\frac{r \cdot \color{blue}{\left(-\sin b\right)}}{-\cos b}\right) \]

       8. distribute-rgt-neg-in 58.9%

          \[\leadsto -1 \cdot \left(-\frac{\color{blue}{-r \cdot \sin b}}{-\cos b}\right) \]

       9. frac-2neg 58.9%

          \[\leadsto -1 \cdot \left(-\color{blue}{\frac{r \cdot \sin b}{\cos b}}\right) \]

       10. associate-/l* 58.8%

           \[\leadsto -1 \cdot \left(-\color{blue}{r \cdot \frac{\sin b}{\cos b}}\right) \]

       11. quot-tan 58.9%

           \[\leadsto -1 \cdot \left(-r \cdot \color{blue}{\tan b}\right) \]

   13. Applied egg-rr 58.9%

       \[\leadsto -1 \cdot \color{blue}{\left(-r \cdot \tan b\right)} \]
   14. Step-by-step derivation

       1. distribute-rgt-neg-in 58.9%

          \[\leadsto -1 \cdot \color{blue}{\left(r \cdot \left(-\tan b\right)\right)} \]

   15. Simplified 58.9%

       \[\leadsto -1 \cdot \color{blue}{\left(r \cdot \left(-\tan b\right)\right)} \]

   if -0.55000000000000004 < b < 1.01999999999999995e-35

   1. Initial program 99.1%

      \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. +-commutative 99.1%

         \[\leadsto r \cdot \frac{\sin b}{\cos \color{blue}{\left(b + a\right)}} \]

   3. Simplified 99.1%

      \[\leadsto \color{blue}{r \cdot \frac{\sin b}{\cos \left(b + a\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. cos-sum 99.8%

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      2. cancel-sign-sub-inv 99.8%

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a + \left(-\sin b\right) \cdot \sin a}} \]

      3. fma-define 99.8%

         \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \sin a\right)}} \]

   6. Applied egg-rr 99.8%

      \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \sin a\right)}} \]

   7. Taylor expanded in b around 0 99.1%

      \[\leadsto \color{blue}{\frac{b \cdot r}{\cos a}} \]
   8. Step-by-step derivation

      1. associate-/l* 99.1%

         \[\leadsto \color{blue}{b \cdot \frac{r}{\cos a}} \]

   9. Simplified 99.1%

      \[\leadsto \color{blue}{b \cdot \frac{r}{\cos a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.55 \lor \neg \left(b \leq 1.02 \cdot 10^{-35}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 50.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ b \cdot \frac{r}{\cos a} \end{array} \]

FPCore

(FPCore (r a b) :precision binary64 (* b (/ r (cos a))))

C

double code(double r, double a, double b) {
	return b * (r / cos(a));
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = b * (r / cos(a))
end function

Java

public static double code(double r, double a, double b) {
	return b * (r / Math.cos(a));
}

Python

def code(r, a, b):
	return b * (r / math.cos(a))

Julia

function code(r, a, b)
	return Float64(b * Float64(r / cos(a)))
end

MATLAB

function tmp = code(r, a, b)
	tmp = b * (r / cos(a));
end

Wolfram

code[r_, a_, b_] := N[(b * N[(r / N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.9%

   \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
2. Step-by-step derivation

   1. +-commutative 78.9%

      \[\leadsto r \cdot \frac{\sin b}{\cos \color{blue}{\left(b + a\right)}} \]

3. Simplified 78.9%

   \[\leadsto \color{blue}{r \cdot \frac{\sin b}{\cos \left(b + a\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. cos-sum 99.5%

      \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

   2. cancel-sign-sub-inv 99.5%

      \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\cos b \cdot \cos a + \left(-\sin b\right) \cdot \sin a}} \]

   3. fma-define 99.5%

      \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \sin a\right)}} \]

6. Applied egg-rr 99.5%

   \[\leadsto r \cdot \frac{\sin b}{\color{blue}{\mathsf{fma}\left(\cos b, \cos a, \left(-\sin b\right) \cdot \sin a\right)}} \]

7. Taylor expanded in b around 0 52.1%

   \[\leadsto \color{blue}{\frac{b \cdot r}{\cos a}} \]
8. Step-by-step derivation

   1. associate-/l* 52.1%

      \[\leadsto \color{blue}{b \cdot \frac{r}{\cos a}} \]

9. Simplified 52.1%

   \[\leadsto \color{blue}{b \cdot \frac{r}{\cos a}} \]

10. Final simplification 52.1%

    \[\leadsto b \cdot \frac{r}{\cos a} \]
11. Add Preprocessing

Alternative 9: 34.8% accurate, 69.0× speedup

Math

\[\begin{array}{l} \\ r \cdot b \end{array} \]

FPCore

(FPCore (r a b) :precision binary64 (* r b))

C

double code(double r, double a, double b) {
	return r * b;
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = r * b
end function

Java

public static double code(double r, double a, double b) {
	return r * b;
}

Python

def code(r, a, b):
	return r * b

Julia

function code(r, a, b)
	return Float64(r * b)
end

MATLAB

function tmp = code(r, a, b)
	tmp = r * b;
end

Wolfram

code[r_, a_, b_] := N[(r * b), $MachinePrecision]

Derivation

1. Initial program 78.9%

   \[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
2. Step-by-step derivation

   1. +-commutative 78.9%

      \[\leadsto r \cdot \frac{\sin b}{\cos \color{blue}{\left(b + a\right)}} \]

3. Simplified 78.9%

   \[\leadsto \color{blue}{r \cdot \frac{\sin b}{\cos \left(b + a\right)}} \]
4. Add Preprocessing

5. Taylor expanded in b around 0 52.1%

   \[\leadsto r \cdot \color{blue}{\frac{b}{\cos a}} \]

6. Taylor expanded in a around 0 36.2%

   \[\leadsto r \cdot \color{blue}{b} \]

7. Final simplification 36.2%

   \[\leadsto r \cdot b \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024079 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))